hypergeometric.tcc

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// Special functions -*- C++ -*-

// Copyright (C) 2006, 2007, 2008, 2009
// Free Software Foundation, Inc.
//
// This file is part of the GNU ISO C++ Library.  This library is free
// software; you can redistribute it and/or modify it under the
// terms of the GNU General Public License as published by the
// Free Software Foundation; either version 3, or (at your option)
// any later version.
//
// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
// GNU General Public License for more details.
//
// Under Section 7 of GPL version 3, you are granted additional
// permissions described in the GCC Runtime Library Exception, version
// 3.1, as published by the Free Software Foundation.

// You should have received a copy of the GNU General Public License and
// a copy of the GCC Runtime Library Exception along with this program;
// see the files COPYING3 and COPYING.RUNTIME respectively.  If not, see
// <http://www.gnu.org/licenses/>.

/** @file tr1/hypergeometric.tcc
 *  This is an internal header file, included by other library headers.
 *  You should not attempt to use it directly.
 */

//
// ISO C++ 14882 TR1: 5.2  Special functions
//

// Written by Edward Smith-Rowland based:
//   (1) Handbook of Mathematical Functions,
//       ed. Milton Abramowitz and Irene A. Stegun,
//       Dover Publications,
//       Section 6, pp. 555-566
//   (2) The Gnu Scientific Library, http://www.gnu.org/software/gsl

#ifndef _GLIBCXX_TR1_HYPERGEOMETRIC_TCC
#define _GLIBCXX_TR1_HYPERGEOMETRIC_TCC 1

namespace std
{
namespace tr1
{

  // [5.2] Special functions

  // Implementation-space details.
  namespace __detail
  {

    /**
     *   @brief This routine returns the confluent hypergeometric function
     *          by series expansion.
     * 
     *   @f[
     *     _1F_1(a;c;x) = \frac{\Gamma(c)}{\Gamma(a)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     * 
     *   If a and b are integers and a < 0 and either b > 0 or b < a then the
     *   series is a polynomial with a finite number of terms.  If b is an integer
     *   and b <= 0 the confluent hypergeometric function is undefined.
     *
     *   @param  __a  The "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    _Tp
    __conf_hyperg_series(const _Tp __a, const _Tp __c, const _Tp __x)
    {
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();

      _Tp __term = _Tp(1);
      _Tp __Fac = _Tp(1);
      const unsigned int __max_iter = 100000;
      unsigned int __i;
      for (__i = 0; __i < __max_iter; ++__i)
        {
          __term *= (__a + _Tp(__i)) * __x
                  / ((__c + _Tp(__i)) * _Tp(1 + __i));
          if (std::abs(__term) < __eps)
            {
              break;
            }
          __Fac += __term;
        }
      if (__i == __max_iter)
        std::__throw_runtime_error(__N("Series failed to converge "
                                       "in __conf_hyperg_series."));

      return __Fac;
    }


    /**
     *  @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *          by an iterative procedure described in
     *          Luke, Algorithms for the Computation of Mathematical Functions.
     *
     *  Like the case of the 2F1 rational approximations, these are 
     *  probably guaranteed to converge for x < 0, barring gross    
     *  numerical instability in the pre-asymptotic regime.         
     */
    template<typename _Tp>
    _Tp
    __conf_hyperg_luke(const _Tp __a, const _Tp __c, const _Tp __xin)
    {
      const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
      const int __nmax = 20000;
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __x  = -__xin;
      const _Tp __x3 = __x * __x * __x;
      const _Tp __t0 = __a / __c;
      const _Tp __t1 = (__a + _Tp(1)) / (_Tp(2) * __c);
      const _Tp __t2 = (__a + _Tp(2)) / (_Tp(2) * (__c + _Tp(1)));
      _Tp __F = _Tp(1);
      _Tp __prec;

      _Tp __Bnm3 = _Tp(1);
      _Tp __Bnm2 = _Tp(1) + __t1 * __x;
      _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);

      _Tp __Anm3 = _Tp(1);
      _Tp __Anm2 = __Bnm2 - __t0 * __x;
      _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
                 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;

      int __n = 3;
      while(1)
        {
          _Tp __npam1 = _Tp(__n - 1) + __a;
          _Tp __npcm1 = _Tp(__n - 1) + __c;
          _Tp __npam2 = _Tp(__n - 2) + __a;
          _Tp __npcm2 = _Tp(__n - 2) + __c;
          _Tp __tnm1  = _Tp(2 * __n - 1);
          _Tp __tnm3  = _Tp(2 * __n - 3);
          _Tp __tnm5  = _Tp(2 * __n - 5);
          _Tp __F1 =  (_Tp(__n - 2) - __a) / (_Tp(2) * __tnm3 * __npcm1);
          _Tp __F2 =  (_Tp(__n) + __a) * __npam1
                   / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
          _Tp __F3 = -__npam2 * __npam1 * (_Tp(__n - 2) - __a)
                   / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
                   * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
          _Tp __E  = -__npam1 * (_Tp(__n - 1) - __c)
                   / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);

          _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
                   + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
          _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
                   + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
          _Tp __r = __An / __Bn;

          __prec = std::abs((__F - __r) / __F);
          __F = __r;

          if (__prec < __eps || __n > __nmax)
            break;

          if (std::abs(__An) > __big || std::abs(__Bn) > __big)
            {
              __An   /= __big;
              __Bn   /= __big;
              __Anm1 /= __big;
              __Bnm1 /= __big;
              __Anm2 /= __big;
              __Bnm2 /= __big;
              __Anm3 /= __big;
              __Bnm3 /= __big;
            }
          else if (std::abs(__An) < _Tp(1) / __big
                || std::abs(__Bn) < _Tp(1) / __big)
            {
              __An   *= __big;
              __Bn   *= __big;
              __Anm1 *= __big;
              __Bnm1 *= __big;
              __Anm2 *= __big;
              __Bnm2 *= __big;
              __Anm3 *= __big;
              __Bnm3 *= __big;
            }

          ++__n;
          __Bnm3 = __Bnm2;
          __Bnm2 = __Bnm1;
          __Bnm1 = __Bn;
          __Anm3 = __Anm2;
          __Anm2 = __Anm1;
          __Anm1 = __An;
        }

      if (__n >= __nmax)
        std::__throw_runtime_error(__N("Iteration failed to converge "
                                       "in __conf_hyperg_luke."));

      return __F;
    }


    /**
     *   @brief  Return the confluent hypogeometric function
     *           @f$ _1F_1(a;c;x) @f$.
     * 
     *   @todo  Handle b == nonpositive integer blowup - return NaN.
     *
     *   @param  __a  The "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    inline _Tp
    __conf_hyperg(const _Tp __a, const _Tp __c, const _Tp __x)
    {
#if _GLIBCXX_USE_C99_MATH_TR1
      const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
      const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
      if (__isnan(__a) || __isnan(__c) || __isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__c_nint == __c && __c_nint <= 0)
        return std::numeric_limits<_Tp>::infinity();
      else if (__a == _Tp(0))
        return _Tp(1);
      else if (__c == __a)
        return std::exp(__x);
      else if (__x < _Tp(0))
        return __conf_hyperg_luke(__a, __c, __x);
      else
        return __conf_hyperg_series(__a, __c, __x);
    }


    /**
     *   @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *   by series expansion.
     * 
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     * 
     *   This works and it's pretty fast.
     *
     *   @param  __a  The first "numerator" parameter.
     *   @param  __a  The second "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    _Tp
    __hyperg_series(const _Tp __a, const _Tp __b,
                    const _Tp __c, const _Tp __x)
    {
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();

      _Tp __term = _Tp(1);
      _Tp __Fabc = _Tp(1);
      const unsigned int __max_iter = 100000;
      unsigned int __i;
      for (__i = 0; __i < __max_iter; ++__i)
        {
          __term *= (__a + _Tp(__i)) * (__b + _Tp(__i)) * __x
                  / ((__c + _Tp(__i)) * _Tp(1 + __i));
          if (std::abs(__term) < __eps)
            {
              break;
            }
          __Fabc += __term;
        }
      if (__i == __max_iter)
        std::__throw_runtime_error(__N("Series failed to converge "
                                       "in __hyperg_series."));

      return __Fabc;
    }


    /**
     *   @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$
     *           by an iterative procedure described in
     *           Luke, Algorithms for the Computation of Mathematical Functions.
     */
    template<typename _Tp>
    _Tp
    __hyperg_luke(const _Tp __a, const _Tp __b, const _Tp __c,
                  const _Tp __xin)
    {
      const _Tp __big = std::pow(std::numeric_limits<_Tp>::max(), _Tp(0.16L));
      const int __nmax = 20000;
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __x  = -__xin;
      const _Tp __x3 = __x * __x * __x;
      const _Tp __t0 = __a * __b / __c;
      const _Tp __t1 = (__a + _Tp(1)) * (__b + _Tp(1)) / (_Tp(2) * __c);
      const _Tp __t2 = (__a + _Tp(2)) * (__b + _Tp(2))
                     / (_Tp(2) * (__c + _Tp(1)));

      _Tp __F = _Tp(1);

      _Tp __Bnm3 = _Tp(1);
      _Tp __Bnm2 = _Tp(1) + __t1 * __x;
      _Tp __Bnm1 = _Tp(1) + __t2 * __x * (_Tp(1) + __t1 / _Tp(3) * __x);

      _Tp __Anm3 = _Tp(1);
      _Tp __Anm2 = __Bnm2 - __t0 * __x;
      _Tp __Anm1 = __Bnm1 - __t0 * (_Tp(1) + __t2 * __x) * __x
                 + __t0 * __t1 * (__c / (__c + _Tp(1))) * __x * __x;

      int __n = 3;
      while (1)
        {
          const _Tp __npam1 = _Tp(__n - 1) + __a;
          const _Tp __npbm1 = _Tp(__n - 1) + __b;
          const _Tp __npcm1 = _Tp(__n - 1) + __c;
          const _Tp __npam2 = _Tp(__n - 2) + __a;
          const _Tp __npbm2 = _Tp(__n - 2) + __b;
          const _Tp __npcm2 = _Tp(__n - 2) + __c;
          const _Tp __tnm1  = _Tp(2 * __n - 1);
          const _Tp __tnm3  = _Tp(2 * __n - 3);
          const _Tp __tnm5  = _Tp(2 * __n - 5);
          const _Tp __n2 = __n * __n;
          const _Tp __F1 = (_Tp(3) * __n2 + (__a + __b - _Tp(6)) * __n
                         + _Tp(2) - __a * __b - _Tp(2) * (__a + __b))
                         / (_Tp(2) * __tnm3 * __npcm1);
          const _Tp __F2 = -(_Tp(3) * __n2 - (__a + __b + _Tp(6)) * __n
                         + _Tp(2) - __a * __b) * __npam1 * __npbm1
                         / (_Tp(4) * __tnm1 * __tnm3 * __npcm2 * __npcm1);
          const _Tp __F3 = (__npam2 * __npam1 * __npbm2 * __npbm1
                         * (_Tp(__n - 2) - __a) * (_Tp(__n - 2) - __b))
                         / (_Tp(8) * __tnm3 * __tnm3 * __tnm5
                         * (_Tp(__n - 3) + __c) * __npcm2 * __npcm1);
          const _Tp __E  = -__npam1 * __npbm1 * (_Tp(__n - 1) - __c)
                         / (_Tp(2) * __tnm3 * __npcm2 * __npcm1);

          _Tp __An = (_Tp(1) + __F1 * __x) * __Anm1
                   + (__E + __F2 * __x) * __x * __Anm2 + __F3 * __x3 * __Anm3;
          _Tp __Bn = (_Tp(1) + __F1 * __x) * __Bnm1
                   + (__E + __F2 * __x) * __x * __Bnm2 + __F3 * __x3 * __Bnm3;
          const _Tp __r = __An / __Bn;

          const _Tp __prec = std::abs((__F - __r) / __F);
          __F = __r;

          if (__prec < __eps || __n > __nmax)
            break;

          if (std::abs(__An) > __big || std::abs(__Bn) > __big)
            {
              __An   /= __big;
              __Bn   /= __big;
              __Anm1 /= __big;
              __Bnm1 /= __big;
              __Anm2 /= __big;
              __Bnm2 /= __big;
              __Anm3 /= __big;
              __Bnm3 /= __big;
            }
          else if (std::abs(__An) < _Tp(1) / __big
                || std::abs(__Bn) < _Tp(1) / __big)
            {
              __An   *= __big;
              __Bn   *= __big;
              __Anm1 *= __big;
              __Bnm1 *= __big;
              __Anm2 *= __big;
              __Bnm2 *= __big;
              __Anm3 *= __big;
              __Bnm3 *= __big;
            }

          ++__n;
          __Bnm3 = __Bnm2;
          __Bnm2 = __Bnm1;
          __Bnm1 = __Bn;
          __Anm3 = __Anm2;
          __Anm2 = __Anm1;
          __Anm1 = __An;
        }

      if (__n >= __nmax)
        std::__throw_runtime_error(__N("Iteration failed to converge "
                                       "in __hyperg_luke."));

      return __F;
    }


    /**
     *  @brief  Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$ by the reflection
     *          formulae in Abramowitz & Stegun formula 15.3.6 for d = c - a - b not integral
     *          and formula 15.3.11 for d = c - a - b integral.
     *          This assumes a, b, c != negative integer.
     *
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     *
     *   The reflection formula for nonintegral @f$ d = c - a - b @f$ is:
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)\Gamma(d)}{\Gamma(c-a)\Gamma(c-b)}
     *                            _2F_1(a,b;1-d;1-x)
     *                    + \frac{\Gamma(c)\Gamma(-d)}{\Gamma(a)\Gamma(b)}
     *                            _2F_1(c-a,c-b;1+d;1-x)
     *   @f]
     *
     *   The reflection formula for integral @f$ m = c - a - b @f$ is:
     *   @f[
     *     _2F_1(a,b;a+b+m;x) = \frac{\Gamma(m)\Gamma(a+b+m)}{\Gamma(a+m)\Gamma(b+m)}
     *                        \sum_{k=0}^{m-1} \frac{(m+a)_k(m+b)_k}{k!(1-m)_k}
     *                      - 
     *   @f]
     */
    template<typename _Tp>
    _Tp
    __hyperg_reflect(const _Tp __a, const _Tp __b, const _Tp __c,
                     const _Tp __x)
    {
      const _Tp __d = __c - __a - __b;
      const int __intd  = std::floor(__d + _Tp(0.5L));
      const _Tp __eps = std::numeric_limits<_Tp>::epsilon();
      const _Tp __toler = _Tp(1000) * __eps;
      const _Tp __log_max = std::log(std::numeric_limits<_Tp>::max());
      const bool __d_integer = (std::abs(__d - __intd) < __toler);

      if (__d_integer)
        {
          const _Tp __ln_omx = std::log(_Tp(1) - __x);
          const _Tp __ad = std::abs(__d);
          _Tp __F1, __F2;

          _Tp __d1, __d2;
          if (__d >= _Tp(0))
            {
              __d1 = __d;
              __d2 = _Tp(0);
            }
          else
            {
              __d1 = _Tp(0);
              __d2 = __d;
            }

          const _Tp __lng_c = __log_gamma(__c);

          //  Evaluate F1.
          if (__ad < __eps)
            {
              //  d = c - a - b = 0.
              __F1 = _Tp(0);
            }
          else
            {

              bool __ok_d1 = true;
              _Tp __lng_ad, __lng_ad1, __lng_bd1;
              __try
                {
                  __lng_ad = __log_gamma(__ad);
                  __lng_ad1 = __log_gamma(__a + __d1);
                  __lng_bd1 = __log_gamma(__b + __d1);
                }
              __catch(...)
                {
                  __ok_d1 = false;
                }

              if (__ok_d1)
                {
                  /* Gamma functions in the denominator are ok.
                   * Proceed with evaluation.
                   */
                  _Tp __sum1 = _Tp(1);
                  _Tp __term = _Tp(1);
                  _Tp __ln_pre1 = __lng_ad + __lng_c + __d2 * __ln_omx
                                - __lng_ad1 - __lng_bd1;

                  /* Do F1 sum.
                   */
                  for (int __i = 1; __i < __ad; ++__i)
                    {
                      const int __j = __i - 1;
                      __term *= (__a + __d2 + __j) * (__b + __d2 + __j)
                              / (_Tp(1) + __d2 + __j) / __i * (_Tp(1) - __x);
                      __sum1 += __term;
                    }

                  if (__ln_pre1 > __log_max)
                    std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                   "in __hyperg_luke."));
                  else
                    __F1 = std::exp(__ln_pre1) * __sum1;
                }
              else
                {
                  //  Gamma functions in the denominator were not ok.
                  //  So the F1 term is zero.
                  __F1 = _Tp(0);
                }
            } // end F1 evaluation

          // Evaluate F2.
          bool __ok_d2 = true;
          _Tp __lng_ad2, __lng_bd2;
          __try
            {
              __lng_ad2 = __log_gamma(__a + __d2);
              __lng_bd2 = __log_gamma(__b + __d2);
            }
          __catch(...)
            {
              __ok_d2 = false;
            }

          if (__ok_d2)
            {
              //  Gamma functions in the denominator are ok.
              //  Proceed with evaluation.
              const int __maxiter = 2000;
              const _Tp __psi_1 = -__numeric_constants<_Tp>::__gamma_e();
              const _Tp __psi_1pd = __psi(_Tp(1) + __ad);
              const _Tp __psi_apd1 = __psi(__a + __d1);
              const _Tp __psi_bpd1 = __psi(__b + __d1);

              _Tp __psi_term = __psi_1 + __psi_1pd - __psi_apd1
                             - __psi_bpd1 - __ln_omx;
              _Tp __fact = _Tp(1);
              _Tp __sum2 = __psi_term;
              _Tp __ln_pre2 = __lng_c + __d1 * __ln_omx
                            - __lng_ad2 - __lng_bd2;

              // Do F2 sum.
              int __j;
              for (__j = 1; __j < __maxiter; ++__j)
                {
                  //  Values for psi functions use recurrence; Abramowitz & Stegun 6.3.5
                  const _Tp __term1 = _Tp(1) / _Tp(__j)
                                    + _Tp(1) / (__ad + __j);
                  const _Tp __term2 = _Tp(1) / (__a + __d1 + _Tp(__j - 1))
                                    + _Tp(1) / (__b + __d1 + _Tp(__j - 1));
                  __psi_term += __term1 - __term2;
                  __fact *= (__a + __d1 + _Tp(__j - 1))
                          * (__b + __d1 + _Tp(__j - 1))
                          / ((__ad + __j) * __j) * (_Tp(1) - __x);
                  const _Tp __delta = __fact * __psi_term;
                  __sum2 += __delta;
                  if (std::abs(__delta) < __eps * std::abs(__sum2))
                    break;
                }
              if (__j == __maxiter)
                std::__throw_runtime_error(__N("Sum F2 failed to converge "
                                               "in __hyperg_reflect"));

              if (__sum2 == _Tp(0))
                __F2 = _Tp(0);
              else
                __F2 = std::exp(__ln_pre2) * __sum2;
            }
          else
            {
              // Gamma functions in the denominator not ok.
              // So the F2 term is zero.
              __F2 = _Tp(0);
            } // end F2 evaluation

          const _Tp __sgn_2 = (__intd % 2 == 1 ? -_Tp(1) : _Tp(1));
          const _Tp __F = __F1 + __sgn_2 * __F2;

          return __F;
        }
      else
        {
          //  d = c - a - b not an integer.

          //  These gamma functions appear in the denominator, so we
          //  catch their harmless domain errors and set the terms to zero.
          bool __ok1 = true;
          _Tp __sgn_g1ca = _Tp(0), __ln_g1ca = _Tp(0);
          _Tp __sgn_g1cb = _Tp(0), __ln_g1cb = _Tp(0);
          __try
            {
              __sgn_g1ca = __log_gamma_sign(__c - __a);
              __ln_g1ca = __log_gamma(__c - __a);
              __sgn_g1cb = __log_gamma_sign(__c - __b);
              __ln_g1cb = __log_gamma(__c - __b);
            }
          __catch(...)
            {
              __ok1 = false;
            }

          bool __ok2 = true;
          _Tp __sgn_g2a = _Tp(0), __ln_g2a = _Tp(0);
          _Tp __sgn_g2b = _Tp(0), __ln_g2b = _Tp(0);
          __try
            {
              __sgn_g2a = __log_gamma_sign(__a);
              __ln_g2a = __log_gamma(__a);
              __sgn_g2b = __log_gamma_sign(__b);
              __ln_g2b = __log_gamma(__b);
            }
          __catch(...)
            {
              __ok2 = false;
            }

          const _Tp __sgn_gc = __log_gamma_sign(__c);
          const _Tp __ln_gc = __log_gamma(__c);
          const _Tp __sgn_gd = __log_gamma_sign(__d);
          const _Tp __ln_gd = __log_gamma(__d);
          const _Tp __sgn_gmd = __log_gamma_sign(-__d);
          const _Tp __ln_gmd = __log_gamma(-__d);

          const _Tp __sgn1 = __sgn_gc * __sgn_gd  * __sgn_g1ca * __sgn_g1cb;
          const _Tp __sgn2 = __sgn_gc * __sgn_gmd * __sgn_g2a  * __sgn_g2b;

          _Tp __pre1, __pre2;
          if (__ok1 && __ok2)
            {
              _Tp __ln_pre1 = __ln_gc + __ln_gd  - __ln_g1ca - __ln_g1cb;
              _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a  - __ln_g2b
                            + __d * std::log(_Tp(1) - __x);
              if (__ln_pre1 < __log_max && __ln_pre2 < __log_max)
                {
                  __pre1 = std::exp(__ln_pre1);
                  __pre2 = std::exp(__ln_pre2);
                  __pre1 *= __sgn1;
                  __pre2 *= __sgn2;
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else if (__ok1 && !__ok2)
            {
              _Tp __ln_pre1 = __ln_gc + __ln_gd - __ln_g1ca - __ln_g1cb;
              if (__ln_pre1 < __log_max)
                {
                  __pre1 = std::exp(__ln_pre1);
                  __pre1 *= __sgn1;
                  __pre2 = _Tp(0);
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else if (!__ok1 && __ok2)
            {
              _Tp __ln_pre2 = __ln_gc + __ln_gmd - __ln_g2a - __ln_g2b
                            + __d * std::log(_Tp(1) - __x);
              if (__ln_pre2 < __log_max)
                {
                  __pre1 = _Tp(0);
                  __pre2 = std::exp(__ln_pre2);
                  __pre2 *= __sgn2;
                }
              else
                {
                  std::__throw_runtime_error(__N("Overflow of gamma functions "
                                                 "in __hyperg_reflect"));
                }
            }
          else
            {
              __pre1 = _Tp(0);
              __pre2 = _Tp(0);
              std::__throw_runtime_error(__N("Underflow of gamma functions "
                                             "in __hyperg_reflect"));
            }

          const _Tp __F1 = __hyperg_series(__a, __b, _Tp(1) - __d,
                                           _Tp(1) - __x);
          const _Tp __F2 = __hyperg_series(__c - __a, __c - __b, _Tp(1) + __d,
                                           _Tp(1) - __x);

          const _Tp __F = __pre1 * __F1 + __pre2 * __F2;

          return __F;
        }
    }


    /**
     *   @brief Return the hypogeometric function @f$ _2F_1(a,b;c;x) @f$.
     *
     *   The hypogeometric function is defined by
     *   @f[
     *     _2F_1(a,b;c;x) = \frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}
     *                      \sum_{n=0}^{\infty}
     *                      \frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)}
     *                      \frac{x^n}{n!}
     *   @f]
     *
     *   @param  __a  The first "numerator" parameter.
     *   @param  __a  The second "numerator" parameter.
     *   @param  __c  The "denominator" parameter.
     *   @param  __x  The argument of the confluent hypergeometric function.
     *   @return  The confluent hypergeometric function.
     */
    template<typename _Tp>
    inline _Tp
    __hyperg(const _Tp __a, const _Tp __b, const _Tp __c, const _Tp __x)
    {
#if _GLIBCXX_USE_C99_MATH_TR1
      const _Tp __a_nint = std::tr1::nearbyint(__a);
      const _Tp __b_nint = std::tr1::nearbyint(__b);
      const _Tp __c_nint = std::tr1::nearbyint(__c);
#else
      const _Tp __a_nint = static_cast<int>(__a + _Tp(0.5L));
      const _Tp __b_nint = static_cast<int>(__b + _Tp(0.5L));
      const _Tp __c_nint = static_cast<int>(__c + _Tp(0.5L));
#endif
      const _Tp __toler = _Tp(1000) * std::numeric_limits<_Tp>::epsilon();
      if (std::abs(__x) >= _Tp(1))
        std::__throw_domain_error(__N("Argument outside unit circle "
                                      "in __hyperg."));
      else if (__isnan(__a) || __isnan(__b)
            || __isnan(__c) || __isnan(__x))
        return std::numeric_limits<_Tp>::quiet_NaN();
      else if (__c_nint == __c && __c_nint <= _Tp(0))
        return std::numeric_limits<_Tp>::infinity();
      else if (std::abs(__c - __b) < __toler || std::abs(__c - __a) < __toler)
        return std::pow(_Tp(1) - __x, __c - __a - __b);
      else if (__a >= _Tp(0) && __b >= _Tp(0) && __c >= _Tp(0)
            && __x >= _Tp(0) && __x < _Tp(0.995L))
        return __hyperg_series(__a, __b, __c, __x);
      else if (std::abs(__a) < _Tp(10) && std::abs(__b) < _Tp(10))
        {
          //  For integer a and b the hypergeometric function is a finite polynomial.
          if (__a < _Tp(0)  &&  std::abs(__a - __a_nint) < __toler)
            return __hyperg_series(__a_nint, __b, __c, __x);
          else if (__b < _Tp(0)  &&  std::abs(__b - __b_nint) < __toler)
            return __hyperg_series(__a, __b_nint, __c, __x);
          else if (__x < -_Tp(0.25L))
            return __hyperg_luke(__a, __b, __c, __x);
          else if (__x < _Tp(0.5L))
            return __hyperg_series(__a, __b, __c, __x);
          else
            if (std::abs(__c) > _Tp(10))
              return __hyperg_series(__a, __b, __c, __x);
            else
              return __hyperg_reflect(__a, __b, __c, __x);
        }
      else
        return __hyperg_luke(__a, __b, __c, __x);
    }

  } // namespace std::tr1::__detail
}
}

#endif // _GLIBCXX_TR1_HYPERGEOMETRIC_TCC